SET007 Axioms: SET007+25.ax
%------------------------------------------------------------------------------
% File : SET007+25 : TPTP v8.2.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : The Modification of a Function by a Function
% Version : [Urb08] axioms.
% English : and the Iteration of the Composition of a Function
% Refs : [Mat90] Matuszewski (1990), Formalized Mathematics
% : [Urb07] Urban (2007), MPTP 0.2: Design, Implementation, and In
% : [Urb08] Urban (2006), Email to G. Sutcliffe
% Source : [Urb08]
% Names : funct_4 [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 95 ( 6 unt; 0 def)
% Number of atoms : 501 ( 86 equ)
% Maximal formula atoms : 14 ( 5 avg)
% Number of connectives : 428 ( 22 ~; 3 |; 204 &)
% ( 12 <=>; 187 =>; 0 <=; 0 <~>)
% Maximal formula depth : 17 ( 7 avg)
% Maximal term depth : 4 ( 1 avg)
% Number of predicates : 13 ( 11 usr; 1 prp; 0-3 aty)
% Number of functors : 20 ( 20 usr; 1 con; 0-5 aty)
% Number of variables : 317 ( 309 !; 8 ?)
% SPC :
% Comments : The individual reference can be found in [Mat90] by looking for
% the name provided by [Urb08].
% : Translated by MPTP from the Mizar Mathematical Library 4.48.930.
% : These set theory axioms are used in encodings of problems in
% various domains, including ALG, CAT, GRP, LAT, SET, and TOP.
%------------------------------------------------------------------------------
fof(fc1_funct_4,axiom,
! [A,B,C,D] :
( v1_relat_1(k4_funct_4(A,B,C,D))
& v1_funct_1(k4_funct_4(A,B,C,D)) ) ).
fof(t1_funct_4,axiom,
! [A] :
~ ( ! [B] :
~ ( r2_hidden(B,A)
& ! [C,D] : B != k4_tarski(C,D) )
& ! [B,C] : ~ r1_tarski(A,k2_zfmisc_1(B,C)) ) ).
fof(t2_funct_4,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A) )
=> ! [B] :
( ( v1_relat_1(B)
& v1_funct_1(B) )
=> k5_relat_1(B,A) = k5_relat_1(B,k7_relat_1(A,k2_relat_1(B))) ) ) ).
fof(t3_funct_4,axiom,
$true ).
fof(t4_funct_4,axiom,
! [A,B] :
( r1_tarski(k6_partfun1(A),k6_partfun1(B))
<=> r1_tarski(A,B) ) ).
fof(t5_funct_4,axiom,
! [A,B,C] :
( r1_tarski(A,B)
=> r1_tarski(k2_funcop_1(A,C),k2_funcop_1(B,C)) ) ).
fof(t6_funct_4,axiom,
! [A,B,C,D] :
( r1_tarski(k2_funcop_1(A,B),k2_funcop_1(C,D))
=> r1_tarski(A,C) ) ).
fof(t7_funct_4,axiom,
! [A,B,C,D] :
( r1_tarski(k2_funcop_1(A,B),k2_funcop_1(C,D))
=> ( A = k1_xboole_0
| B = D ) ) ).
fof(t8_funct_4,axiom,
! [A,B] :
( ( v1_relat_1(B)
& v1_funct_1(B) )
=> ( r2_hidden(A,k1_relat_1(B))
=> r1_tarski(k2_funcop_1(k1_tarski(A),k1_funct_1(B,A)),B) ) ) ).
fof(t9_funct_4,axiom,
! [A,B,C] :
( ( v1_relat_1(C)
& v1_funct_1(C) )
=> r1_tarski(k7_relat_1(k8_relat_1(A,C),B),C) ) ).
fof(t10_funct_4,axiom,
! [A,B,C] :
( ( v1_relat_1(C)
& v1_funct_1(C) )
=> ! [D] :
( ( v1_relat_1(D)
& v1_funct_1(D) )
=> ( r1_tarski(C,D)
=> r1_tarski(k7_relat_1(k8_relat_1(A,C),B),k7_relat_1(k8_relat_1(A,D),B)) ) ) ) ).
fof(d1_funct_4,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A) )
=> ! [B] :
( ( v1_relat_1(B)
& v1_funct_1(B) )
=> ! [C] :
( ( v1_relat_1(C)
& v1_funct_1(C) )
=> ( C = k1_funct_4(A,B)
<=> ( k1_relat_1(C) = k2_xboole_0(k1_relat_1(A),k1_relat_1(B))
& ! [D] :
( r2_hidden(D,k2_xboole_0(k1_relat_1(A),k1_relat_1(B)))
=> ( ( r2_hidden(D,k1_relat_1(B))
=> k1_funct_1(C,D) = k1_funct_1(B,D) )
& ( ~ r2_hidden(D,k1_relat_1(B))
=> k1_funct_1(C,D) = k1_funct_1(A,D) ) ) ) ) ) ) ) ) ).
fof(t11_funct_4,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A) )
=> ! [B] :
( ( v1_relat_1(B)
& v1_funct_1(B) )
=> ( r1_tarski(k1_relat_1(A),k1_relat_1(k1_funct_4(A,B)))
& r1_tarski(k1_relat_1(B),k1_relat_1(k1_funct_4(A,B))) ) ) ) ).
fof(t12_funct_4,axiom,
! [A,B] :
( ( v1_relat_1(B)
& v1_funct_1(B) )
=> ! [C] :
( ( v1_relat_1(C)
& v1_funct_1(C) )
=> ( ~ r2_hidden(A,k1_relat_1(B))
=> k1_funct_1(k1_funct_4(C,B),A) = k1_funct_1(C,A) ) ) ) ).
fof(t13_funct_4,axiom,
! [A,B] :
( ( v1_relat_1(B)
& v1_funct_1(B) )
=> ! [C] :
( ( v1_relat_1(C)
& v1_funct_1(C) )
=> ( r2_hidden(A,k1_relat_1(k1_funct_4(B,C)))
<=> ( r2_hidden(A,k1_relat_1(B))
| r2_hidden(A,k1_relat_1(C)) ) ) ) ) ).
fof(t14_funct_4,axiom,
! [A,B] :
( ( v1_relat_1(B)
& v1_funct_1(B) )
=> ! [C] :
( ( v1_relat_1(C)
& v1_funct_1(C) )
=> ( r2_hidden(A,k1_relat_1(B))
=> k1_funct_1(k1_funct_4(C,B),A) = k1_funct_1(B,A) ) ) ) ).
fof(t15_funct_4,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A) )
=> ! [B] :
( ( v1_relat_1(B)
& v1_funct_1(B) )
=> ! [C] :
( ( v1_relat_1(C)
& v1_funct_1(C) )
=> k1_funct_4(k1_funct_4(A,B),C) = k1_funct_4(A,k1_funct_4(B,C)) ) ) ) ).
fof(t16_funct_4,axiom,
! [A,B] :
( ( v1_relat_1(B)
& v1_funct_1(B) )
=> ! [C] :
( ( v1_relat_1(C)
& v1_funct_1(C) )
=> ( ( r1_partfun1(B,C)
& r2_hidden(A,k1_relat_1(B)) )
=> k1_funct_1(k1_funct_4(B,C),A) = k1_funct_1(B,A) ) ) ) ).
fof(t17_funct_4,axiom,
! [A,B] :
( ( v1_relat_1(B)
& v1_funct_1(B) )
=> ! [C] :
( ( v1_relat_1(C)
& v1_funct_1(C) )
=> ( ( r1_xboole_0(k1_relat_1(B),k1_relat_1(C))
& r2_hidden(A,k1_relat_1(B)) )
=> k1_funct_1(k1_funct_4(B,C),A) = k1_funct_1(B,A) ) ) ) ).
fof(t18_funct_4,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A) )
=> ! [B] :
( ( v1_relat_1(B)
& v1_funct_1(B) )
=> r1_tarski(k2_relat_1(k1_funct_4(A,B)),k2_xboole_0(k2_relat_1(A),k2_relat_1(B))) ) ) ).
fof(t19_funct_4,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A) )
=> ! [B] :
( ( v1_relat_1(B)
& v1_funct_1(B) )
=> r1_tarski(k2_relat_1(A),k2_relat_1(k1_funct_4(B,A))) ) ) ).
fof(t20_funct_4,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A) )
=> ! [B] :
( ( v1_relat_1(B)
& v1_funct_1(B) )
=> ( r1_tarski(k1_relat_1(A),k1_relat_1(B))
=> k1_funct_4(A,B) = B ) ) ) ).
fof(t21_funct_4,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A) )
=> k1_funct_4(k1_xboole_0,A) = A ) ).
fof(t22_funct_4,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A) )
=> k1_funct_4(A,k1_xboole_0) = A ) ).
fof(t23_funct_4,axiom,
! [A,B] : k1_funct_4(k6_partfun1(A),k6_partfun1(B)) = k6_partfun1(k2_xboole_0(A,B)) ).
fof(t24_funct_4,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A) )
=> ! [B] :
( ( v1_relat_1(B)
& v1_funct_1(B) )
=> k7_relat_1(k1_funct_4(A,B),k1_relat_1(B)) = B ) ) ).
fof(t25_funct_4,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A) )
=> ! [B] :
( ( v1_relat_1(B)
& v1_funct_1(B) )
=> r1_tarski(k7_relat_1(k1_funct_4(A,B),k4_xboole_0(k1_relat_1(A),k1_relat_1(B))),A) ) ) ).
fof(t26_funct_4,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A) )
=> ! [B] :
( ( v1_relat_1(B)
& v1_funct_1(B) )
=> r1_tarski(A,k1_funct_4(B,A)) ) ) ).
fof(t27_funct_4,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A) )
=> ! [B] :
( ( v1_relat_1(B)
& v1_funct_1(B) )
=> ! [C] :
( ( v1_relat_1(C)
& v1_funct_1(C) )
=> ( r1_partfun1(A,k1_funct_4(B,C))
=> r1_partfun1(k7_relat_1(A,k4_xboole_0(k1_relat_1(A),k1_relat_1(C))),B) ) ) ) ) ).
fof(t28_funct_4,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A) )
=> ! [B] :
( ( v1_relat_1(B)
& v1_funct_1(B) )
=> ! [C] :
( ( v1_relat_1(C)
& v1_funct_1(C) )
=> ( r1_partfun1(A,k1_funct_4(B,C))
=> r1_partfun1(A,C) ) ) ) ) ).
fof(t29_funct_4,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A) )
=> ! [B] :
( ( v1_relat_1(B)
& v1_funct_1(B) )
=> ( r1_partfun1(A,B)
<=> r1_tarski(A,k1_funct_4(A,B)) ) ) ) ).
fof(t30_funct_4,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A) )
=> ! [B] :
( ( v1_relat_1(B)
& v1_funct_1(B) )
=> r1_tarski(k1_funct_4(A,B),k2_xboole_0(A,B)) ) ) ).
fof(t31_funct_4,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A) )
=> ! [B] :
( ( v1_relat_1(B)
& v1_funct_1(B) )
=> ( r1_partfun1(A,B)
<=> k2_xboole_0(A,B) = k1_funct_4(A,B) ) ) ) ).
fof(t32_funct_4,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A) )
=> ! [B] :
( ( v1_relat_1(B)
& v1_funct_1(B) )
=> ( r1_xboole_0(k1_relat_1(A),k1_relat_1(B))
=> k2_xboole_0(A,B) = k1_funct_4(A,B) ) ) ) ).
fof(t33_funct_4,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A) )
=> ! [B] :
( ( v1_relat_1(B)
& v1_funct_1(B) )
=> ( r1_xboole_0(k1_relat_1(A),k1_relat_1(B))
=> r1_tarski(A,k1_funct_4(A,B)) ) ) ) ).
fof(t34_funct_4,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A) )
=> ! [B] :
( ( v1_relat_1(B)
& v1_funct_1(B) )
=> ( r1_xboole_0(k1_relat_1(A),k1_relat_1(B))
=> k7_relat_1(k1_funct_4(A,B),k1_relat_1(A)) = A ) ) ) ).
fof(t35_funct_4,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A) )
=> ! [B] :
( ( v1_relat_1(B)
& v1_funct_1(B) )
=> ( r1_partfun1(A,B)
<=> k1_funct_4(A,B) = k1_funct_4(B,A) ) ) ) ).
fof(t36_funct_4,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A) )
=> ! [B] :
( ( v1_relat_1(B)
& v1_funct_1(B) )
=> ( r1_xboole_0(k1_relat_1(A),k1_relat_1(B))
=> k1_funct_4(A,B) = k1_funct_4(B,A) ) ) ) ).
fof(t37_funct_4,axiom,
! [A,B,C] :
( ( v1_funct_1(C)
& m2_relset_1(C,A,B) )
=> ! [D] :
( ( v1_funct_1(D)
& m2_relset_1(D,A,B) )
=> ( v1_partfun1(D,A,B)
=> k1_funct_4(C,D) = D ) ) ) ).
fof(t38_funct_4,axiom,
! [A,B,C] :
( ( v1_funct_1(C)
& v1_funct_2(C,A,B)
& m2_relset_1(C,A,B) )
=> ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,A,B)
& m2_relset_1(D,A,B) )
=> ( ( B = k1_xboole_0
=> A = k1_xboole_0 )
=> k1_funct_4(C,D) = D ) ) ) ).
fof(t39_funct_4,axiom,
! [A,B] :
( ( v1_funct_1(B)
& v1_funct_2(B,A,A)
& m2_relset_1(B,A,A) )
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,A,A)
& m2_relset_1(C,A,A) )
=> k1_funct_4(B,C) = C ) ) ).
fof(t40_funct_4,axiom,
! [A,B] :
( ~ v1_xboole_0(B)
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,A,B)
& m2_relset_1(C,A,B) )
=> ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,A,B)
& m2_relset_1(D,A,B) )
=> k1_funct_4(C,D) = D ) ) ) ).
fof(t41_funct_4,axiom,
! [A,B,C] :
( ( v1_funct_1(C)
& m2_relset_1(C,A,B) )
=> ! [D] :
( ( v1_funct_1(D)
& m2_relset_1(D,A,B) )
=> ( v1_funct_1(k1_funct_4(C,D))
& m2_relset_1(k1_funct_4(C,D),A,B) ) ) ) ).
fof(d2_funct_4,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A) )
=> ! [B] :
( ( v1_relat_1(B)
& v1_funct_1(B) )
=> ( B = k2_funct_4(A)
<=> ( ! [C] :
( r2_hidden(C,k1_relat_1(B))
<=> ? [D,E] :
( C = k4_tarski(E,D)
& r2_hidden(k4_tarski(D,E),k1_relat_1(A)) ) )
& ! [C,D] :
( r2_hidden(k4_tarski(C,D),k1_relat_1(A))
=> k1_funct_1(B,k4_tarski(D,C)) = k1_funct_1(A,k4_tarski(C,D)) ) ) ) ) ) ).
fof(t42_funct_4,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A) )
=> r1_tarski(k2_relat_1(k2_funct_4(A)),k2_relat_1(A)) ) ).
fof(t43_funct_4,axiom,
! [A,B,C] :
( ( v1_relat_1(C)
& v1_funct_1(C) )
=> ( r2_hidden(k4_tarski(A,B),k1_relat_1(C))
<=> r2_hidden(k4_tarski(B,A),k1_relat_1(k2_funct_4(C))) ) ) ).
fof(t44_funct_4,axiom,
! [A,B,C] :
( ( v1_relat_1(C)
& v1_funct_1(C) )
=> ( r2_hidden(k4_tarski(A,B),k1_relat_1(k2_funct_4(C)))
=> k1_funct_1(k2_funct_4(C),k4_tarski(A,B)) = k1_funct_1(C,k4_tarski(B,A)) ) ) ).
fof(t45_funct_4,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A) )
=> ? [B,C] : r1_tarski(k1_relat_1(k2_funct_4(A)),k2_zfmisc_1(B,C)) ) ).
fof(t46_funct_4,axiom,
! [A,B,C] :
( ( v1_relat_1(C)
& v1_funct_1(C) )
=> ( r1_tarski(k1_relat_1(C),k2_zfmisc_1(A,B))
=> r1_tarski(k1_relat_1(k2_funct_4(C)),k2_zfmisc_1(B,A)) ) ) ).
fof(t47_funct_4,axiom,
! [A,B,C] :
( ( v1_relat_1(C)
& v1_funct_1(C) )
=> ( k1_relat_1(C) = k2_zfmisc_1(A,B)
=> k1_relat_1(k2_funct_4(C)) = k2_zfmisc_1(B,A) ) ) ).
fof(t48_funct_4,axiom,
! [A,B,C] :
( ( v1_relat_1(C)
& v1_funct_1(C) )
=> ( r1_tarski(k1_relat_1(C),k2_zfmisc_1(A,B))
=> k2_relat_1(k2_funct_4(C)) = k2_relat_1(C) ) ) ).
fof(t49_funct_4,axiom,
! [A,B,C,D] :
( ( v1_funct_1(D)
& m2_relset_1(D,k2_zfmisc_1(A,B),C) )
=> ( v1_funct_1(k2_funct_4(D))
& m2_relset_1(k2_funct_4(D),k2_zfmisc_1(B,A),C) ) ) ).
fof(t50_funct_4,axiom,
! [A,B,C,D] :
( ( v1_funct_1(D)
& v1_funct_2(D,k2_zfmisc_1(A,B),C)
& m2_relset_1(D,k2_zfmisc_1(A,B),C) )
=> ( C != k1_xboole_0
=> ( v1_funct_1(k2_funct_4(D))
& v1_funct_2(k2_funct_4(D),k2_zfmisc_1(B,A),C)
& m2_relset_1(k2_funct_4(D),k2_zfmisc_1(B,A),C) ) ) ) ).
fof(t51_funct_4,axiom,
! [A,B,C] :
( ~ v1_xboole_0(C)
=> ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,k2_zfmisc_1(A,B),C)
& m2_relset_1(D,k2_zfmisc_1(A,B),C) )
=> ( v1_funct_1(k2_funct_4(D))
& v1_funct_2(k2_funct_4(D),k2_zfmisc_1(B,A),C)
& m2_relset_1(k2_funct_4(D),k2_zfmisc_1(B,A),C) ) ) ) ).
fof(t52_funct_4,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A) )
=> r1_tarski(k2_funct_4(k2_funct_4(A)),A) ) ).
fof(t53_funct_4,axiom,
! [A,B,C] :
( ( v1_relat_1(C)
& v1_funct_1(C) )
=> ( r1_tarski(k1_relat_1(C),k2_zfmisc_1(A,B))
=> k2_funct_4(k2_funct_4(C)) = C ) ) ).
fof(t54_funct_4,axiom,
! [A,B,C,D] :
( ( v1_funct_1(D)
& m2_relset_1(D,k2_zfmisc_1(A,B),C) )
=> k2_funct_4(k2_funct_4(D)) = D ) ).
fof(t55_funct_4,axiom,
! [A,B,C,D] :
( ( v1_funct_1(D)
& v1_funct_2(D,k2_zfmisc_1(A,B),C)
& m2_relset_1(D,k2_zfmisc_1(A,B),C) )
=> ( C != k1_xboole_0
=> k2_funct_4(k2_funct_4(D)) = D ) ) ).
fof(t56_funct_4,axiom,
! [A,B,C] :
( ~ v1_xboole_0(C)
=> ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,k2_zfmisc_1(A,B),C)
& m2_relset_1(D,k2_zfmisc_1(A,B),C) )
=> k2_funct_4(k2_funct_4(D)) = D ) ) ).
fof(d3_funct_4,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A) )
=> ! [B] :
( ( v1_relat_1(B)
& v1_funct_1(B) )
=> ! [C] :
( ( v1_relat_1(C)
& v1_funct_1(C) )
=> ( C = k3_funct_4(A,B)
<=> ( ! [D] :
( r2_hidden(D,k1_relat_1(C))
<=> ? [E,F,G,H] :
( D = k4_tarski(k4_tarski(E,G),k4_tarski(F,H))
& r2_hidden(k4_tarski(E,F),k1_relat_1(A))
& r2_hidden(k4_tarski(G,H),k1_relat_1(B)) ) )
& ! [D,E,F,G] :
( ( r2_hidden(k4_tarski(D,E),k1_relat_1(A))
& r2_hidden(k4_tarski(F,G),k1_relat_1(B)) )
=> k1_funct_1(C,k4_tarski(k4_tarski(D,F),k4_tarski(E,G))) = k4_tarski(k1_funct_1(A,k4_tarski(D,E)),k1_funct_1(B,k4_tarski(F,G))) ) ) ) ) ) ) ).
fof(t57_funct_4,axiom,
! [A,B,C,D,E] :
( ( v1_relat_1(E)
& v1_funct_1(E) )
=> ! [F] :
( ( v1_relat_1(F)
& v1_funct_1(F) )
=> ( r2_hidden(k4_tarski(k4_tarski(A,B),k4_tarski(C,D)),k1_relat_1(k3_funct_4(E,F)))
<=> ( r2_hidden(k4_tarski(A,C),k1_relat_1(E))
& r2_hidden(k4_tarski(B,D),k1_relat_1(F)) ) ) ) ) ).
fof(t58_funct_4,axiom,
! [A,B,C,D,E] :
( ( v1_relat_1(E)
& v1_funct_1(E) )
=> ! [F] :
( ( v1_relat_1(F)
& v1_funct_1(F) )
=> ( r2_hidden(k4_tarski(k4_tarski(A,B),k4_tarski(C,D)),k1_relat_1(k3_funct_4(E,F)))
=> k1_funct_1(k3_funct_4(E,F),k4_tarski(k4_tarski(A,B),k4_tarski(C,D))) = k4_tarski(k1_funct_1(E,k4_tarski(A,C)),k1_funct_1(F,k4_tarski(B,D))) ) ) ) ).
fof(t59_funct_4,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A) )
=> ! [B] :
( ( v1_relat_1(B)
& v1_funct_1(B) )
=> r1_tarski(k2_relat_1(k3_funct_4(A,B)),k2_zfmisc_1(k2_relat_1(A),k2_relat_1(B))) ) ) ).
fof(t60_funct_4,axiom,
! [A,B,C,D,E] :
( ( v1_relat_1(E)
& v1_funct_1(E) )
=> ! [F] :
( ( v1_relat_1(F)
& v1_funct_1(F) )
=> ( ( r1_tarski(k1_relat_1(E),k2_zfmisc_1(A,B))
& r1_tarski(k1_relat_1(F),k2_zfmisc_1(C,D)) )
=> r1_tarski(k1_relat_1(k3_funct_4(E,F)),k2_zfmisc_1(k2_zfmisc_1(A,C),k2_zfmisc_1(B,D))) ) ) ) ).
fof(t61_funct_4,axiom,
! [A,B,C,D,E] :
( ( v1_relat_1(E)
& v1_funct_1(E) )
=> ! [F] :
( ( v1_relat_1(F)
& v1_funct_1(F) )
=> ( ( k1_relat_1(E) = k2_zfmisc_1(A,B)
& k1_relat_1(F) = k2_zfmisc_1(C,D) )
=> k1_relat_1(k3_funct_4(E,F)) = k2_zfmisc_1(k2_zfmisc_1(A,C),k2_zfmisc_1(B,D)) ) ) ) ).
fof(t62_funct_4,axiom,
! [A,B,C,D,E,F,G] :
( ( v1_funct_1(G)
& m2_relset_1(G,k2_zfmisc_1(A,B),C) )
=> ! [H] :
( ( v1_funct_1(H)
& m2_relset_1(H,k2_zfmisc_1(D,E),F) )
=> ( v1_funct_1(k3_funct_4(G,H))
& m2_relset_1(k3_funct_4(G,H),k2_zfmisc_1(k2_zfmisc_1(A,D),k2_zfmisc_1(B,E)),k2_zfmisc_1(C,F)) ) ) ) ).
fof(t63_funct_4,axiom,
! [A,B,C,D,E,F,G] :
( ( v1_funct_1(G)
& v1_funct_2(G,k2_zfmisc_1(A,B),C)
& m2_relset_1(G,k2_zfmisc_1(A,B),C) )
=> ! [H] :
( ( v1_funct_1(H)
& v1_funct_2(H,k2_zfmisc_1(D,E),F)
& m2_relset_1(H,k2_zfmisc_1(D,E),F) )
=> ~ ( C != k1_xboole_0
& F != k1_xboole_0
& ~ ( v1_funct_1(k3_funct_4(G,H))
& v1_funct_2(k3_funct_4(G,H),k2_zfmisc_1(k2_zfmisc_1(A,D),k2_zfmisc_1(B,E)),k2_zfmisc_1(C,F))
& m2_relset_1(k3_funct_4(G,H),k2_zfmisc_1(k2_zfmisc_1(A,D),k2_zfmisc_1(B,E)),k2_zfmisc_1(C,F)) ) ) ) ) ).
fof(t64_funct_4,axiom,
! [A,B,C,D,E] :
( ~ v1_xboole_0(E)
=> ! [F] :
( ~ v1_xboole_0(F)
=> ! [G] :
( ( v1_funct_1(G)
& v1_funct_2(G,k2_zfmisc_1(A,B),E)
& m2_relset_1(G,k2_zfmisc_1(A,B),E) )
=> ! [H] :
( ( v1_funct_1(H)
& v1_funct_2(H,k2_zfmisc_1(C,D),F)
& m2_relset_1(H,k2_zfmisc_1(C,D),F) )
=> ( v1_funct_1(k3_funct_4(G,H))
& v1_funct_2(k3_funct_4(G,H),k2_zfmisc_1(k2_zfmisc_1(A,C),k2_zfmisc_1(B,D)),k2_zfmisc_1(E,F))
& m2_relset_1(k3_funct_4(G,H),k2_zfmisc_1(k2_zfmisc_1(A,C),k2_zfmisc_1(B,D)),k2_zfmisc_1(E,F)) ) ) ) ) ) ).
fof(d4_funct_4,axiom,
! [A,B,C,D] : k4_funct_4(A,B,C,D) = k1_funct_4(k2_funcop_1(k1_tarski(A),C),k2_funcop_1(k1_tarski(B),D)) ).
fof(t65_funct_4,axiom,
! [A,B,C,D] :
( k1_relat_1(k4_funct_4(A,B,C,D)) = k2_tarski(A,B)
& r1_tarski(k2_relat_1(k4_funct_4(A,B,C,D)),k2_tarski(C,D)) ) ).
fof(t66_funct_4,axiom,
! [A,B,C,D] :
( A != B
=> ( k1_funct_1(k4_funct_4(A,B,C,D),A) = C
& k1_funct_1(k4_funct_4(A,B,C,D),B) = D ) ) ).
fof(t67_funct_4,axiom,
! [A,B,C,D] :
( A != B
=> k2_relat_1(k4_funct_4(A,B,C,D)) = k2_tarski(C,D) ) ).
fof(t68_funct_4,axiom,
! [A,B,C] : k4_funct_4(A,B,C,C) = k2_funcop_1(k2_tarski(A,B),C) ).
fof(t69_funct_4,axiom,
! [A,B,C,D,E] :
( ( v1_relat_1(E)
& v1_funct_1(E) )
=> ( ( k1_relat_1(E) = k2_tarski(A,B)
& k1_funct_1(E,A) = C
& k1_funct_1(E,B) = D )
=> E = k4_funct_4(A,B,C,D) ) ) ).
fof(t70_funct_4,axiom,
! [A,B] : k2_funcop_1(k1_tarski(A),B) = k1_tarski(k4_tarski(A,B)) ).
fof(t71_funct_4,axiom,
! [A,B,C,D] :
( A != C
=> k4_funct_4(A,C,B,D) = k2_tarski(k4_tarski(A,B),k4_tarski(C,D)) ) ).
fof(t72_funct_4,axiom,
! [A,B,C,D,E,F] :
( k4_funct_4(A,B,C,D) = k4_funct_4(A,B,E,F)
=> ( A = B
| ( C = E
& D = F ) ) ) ).
fof(t73_funct_4,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A) )
=> ! [B] :
( ( v1_relat_1(B)
& v1_funct_1(B) )
=> ! [C] :
( ( v1_relat_1(C)
& v1_funct_1(C) )
=> ! [D] :
( ( v1_relat_1(D)
& v1_funct_1(D) )
=> ( ( r1_tarski(k2_relat_1(C),k1_relat_1(A))
& r1_tarski(k2_relat_1(D),k1_relat_1(B))
& r1_partfun1(A,B) )
=> k5_relat_1(k1_funct_4(C,D),k1_funct_4(A,B)) = k1_funct_4(k5_relat_1(C,A),k5_relat_1(D,B)) ) ) ) ) ) ).
fof(t74_funct_4,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A) )
=> ! [B,C] :
( r1_tarski(k1_relat_1(A),k2_xboole_0(B,C))
=> k1_funct_4(k7_relat_1(A,B),k7_relat_1(A,C)) = A ) ) ).
fof(t75_funct_4,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A) )
=> ! [B] :
( ( v1_relat_1(B)
& v1_funct_1(B) )
=> ! [C] : k7_relat_1(k1_funct_4(A,B),C) = k1_funct_4(k7_relat_1(A,C),k7_relat_1(B,C)) ) ) ).
fof(t76_funct_4,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A) )
=> ! [B] :
( ( v1_relat_1(B)
& v1_funct_1(B) )
=> ! [C] :
( r1_xboole_0(C,k1_relat_1(B))
=> k7_relat_1(k1_funct_4(A,B),C) = k7_relat_1(A,C) ) ) ) ).
fof(t77_funct_4,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A) )
=> ! [B] :
( ( v1_relat_1(B)
& v1_funct_1(B) )
=> ! [C] :
( r1_xboole_0(k1_relat_1(A),C)
=> k7_relat_1(k1_funct_4(A,B),C) = k7_relat_1(B,C) ) ) ) ).
fof(t78_funct_4,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A) )
=> ! [B] :
( ( v1_relat_1(B)
& v1_funct_1(B) )
=> ! [C] :
( ( v1_relat_1(C)
& v1_funct_1(C) )
=> ( k1_relat_1(B) = k1_relat_1(C)
=> k1_funct_4(k1_funct_4(A,B),C) = k1_funct_4(A,C) ) ) ) ) ).
fof(t79_funct_4,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A) )
=> ! [B] :
( ( v1_relat_1(B)
& v1_funct_1(B) )
=> ( r1_tarski(A,B)
=> ( k1_funct_4(A,B) = B
& k1_funct_4(B,A) = B ) ) ) ) ).
fof(t80_funct_4,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A) )
=> ! [B] : k1_funct_4(A,k7_relat_1(A,B)) = A ) ).
fof(t81_funct_4,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A) )
=> ! [B] :
( ( v1_relat_1(B)
& v1_funct_1(B) )
=> ! [C,D] :
( ( r1_tarski(k1_relat_1(A),C)
& r1_tarski(k1_relat_1(B),D)
& r1_xboole_0(C,D) )
=> ( k7_relat_1(k1_funct_4(A,B),C) = A
& k7_relat_1(k1_funct_4(A,B),D) = B ) ) ) ) ).
fof(t82_funct_4,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A) )
=> ! [B] :
( ( v1_relat_1(B)
& v1_funct_1(B) )
=> ! [C] :
( ( r1_tarski(k1_relat_1(A),C)
& r1_xboole_0(k1_relat_1(B),C) )
=> k7_relat_1(k1_funct_4(A,B),C) = A ) ) ) ).
fof(t83_funct_4,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A) )
=> ! [B,C] : k7_relat_1(A,k2_xboole_0(B,C)) = k1_funct_4(k7_relat_1(A,B),k7_relat_1(A,C)) ) ).
fof(dt_k1_funct_4,axiom,
! [A,B] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v1_relat_1(B)
& v1_funct_1(B) )
=> ( v1_relat_1(k1_funct_4(A,B))
& v1_funct_1(k1_funct_4(A,B)) ) ) ).
fof(idempotence_k1_funct_4,axiom,
! [A,B] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v1_relat_1(B)
& v1_funct_1(B) )
=> k1_funct_4(A,A) = A ) ).
fof(dt_k2_funct_4,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A) )
=> ( v1_relat_1(k2_funct_4(A))
& v1_funct_1(k2_funct_4(A)) ) ) ).
fof(dt_k3_funct_4,axiom,
! [A,B] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v1_relat_1(B)
& v1_funct_1(B) )
=> ( v1_relat_1(k3_funct_4(A,B))
& v1_funct_1(k3_funct_4(A,B)) ) ) ).
fof(dt_k4_funct_4,axiom,
$true ).
fof(dt_k5_funct_4,axiom,
! [A,B,C,D,E] :
( ( ~ v1_xboole_0(A)
& m1_subset_1(D,A)
& m1_subset_1(E,A) )
=> ( v1_funct_1(k5_funct_4(A,B,C,D,E))
& v1_funct_2(k5_funct_4(A,B,C,D,E),k2_tarski(B,C),A)
& m2_relset_1(k5_funct_4(A,B,C,D,E),k2_tarski(B,C),A) ) ) ).
fof(redefinition_k5_funct_4,axiom,
! [A,B,C,D,E] :
( ( ~ v1_xboole_0(A)
& m1_subset_1(D,A)
& m1_subset_1(E,A) )
=> k5_funct_4(A,B,C,D,E) = k4_funct_4(B,C,D,E) ) ).
%------------------------------------------------------------------------------